1
JEE Main 2021 (Online) 20th July Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language
Let $$A = \{ {a_{ij}}\} $$ be a 3 $$\times$$ 3 matrix,

where $${a_{ij}} = \left\{ {\matrix{ {{{( - 1)}^{j - i}}} & {if} & {i < j,} \cr 2 & {if} & {i = j,} \cr {{{( - 1)}^{i + j}}} & {if} & {i > j} \cr } } \right.$$

then $$\det (3Adj(2{A^{ - 1}}))$$ is equal to _____________.
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2
JEE Main 2021 (Online) 20th July Evening Shift
Numerical
+4
-1
Change Language
The number of solutions of the equation

$${\log _{(x + 1)}}(2{x^2} + 7x + 5) + {\log _{(2x + 5)}}{(x + 1)^2} - 4 = 0$$, x > 0, is :
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3
JEE Main 2021 (Online) 20th July Evening Shift
Numerical
+4
-1
Change Language
Let a curve y = y(x) be given by the solution of the differential equation $$\cos \left( {{1 \over 2}{{\cos }^{ - 1}}({e^{ - x}})} \right)dx = \sqrt {{e^{2x}} - 1} dy$$. If it intersects y-axis at y = $$-$$1, and the intersection point of the curve with x-axis is ($$\alpha$$, 0), then e$$\alpha$$ is equal to __________________.
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4
JEE Main 2021 (Online) 20th July Evening Shift
Numerical
+4
-1
Change Language
For p > 0, a vector $${\overrightarrow v _2} = 2\widehat i + (p + 1)\widehat j$$ is obtained by rotating the vector $${\overrightarrow v _1} = \sqrt 3 p\widehat i + \widehat j$$ by an angle $$\theta$$ about origin in counter clockwise direction. If $$\tan \theta = {{\left( {\alpha \sqrt 3 - 2} \right)} \over {\left( {4\sqrt 3 + 3} \right)}}$$, then the value of $$\alpha$$ is equal to _____________.
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